Inequalities of quotients of elementary symmetric polynomials

Question

Many inequalities regarding symmetric polynomials such as this are posed as problems

• http://www.artofproblemsolving.com/Wiki/index.php/1997_USAMO_Problems/Problem_5 $$(a^3+b^3+abc)^{-1}+(b^3+c^3+abc)^{-1}+(a^3+c^3+abc)^{-1}\le(abc)^{-1}$$
• http://www.artofproblemsolving.com/Wiki/index.php/2003_USAMO_Problems/Problem_5 $$\frac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \frac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \frac{(2c + a + b)^2}{2c^2 + (a + b)^2} \le 8$$
• http://www.artofproblemsolving.com/Wiki/index.php/2004_USAMO_Problems/Problem_5 $$(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \ge (a+b+c)^3$$
• http://www.artofproblemsolving.com/Wiki/index.php/1980_USAMO_Problems/Problem_5 $$\frac{x}{y + z + 1} + \frac{y}{z + x + 1} + \frac{z}{x + y + 1} \le 1 - (1 - x)(1 - y)(1 - z)$$

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Is there an uniform method or algorithm to prove all true ones? We can assume they are input in some basis for symmetric polynomials.

Answer 1

There is work done in this regard and some interesting results. I recommend the following articles:

• Cubic And Quartic Cyclic Homogenous Inequalities of Three Variables
• Inequalities for Symmetric Means
• Necessary and sucient conditions for symmetric homogeneous polynomial inequalities of degree four and six in real variables

It is a fascinating area of research in which a lot remains to be discovered.